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MIT 6.441. Capacity of multi-antenna Gaussian Channels, I. E. Telatar. May 11, 2006. By: Imad Jabbour. Introduction. MIMO systems in wireless comm. Recently subject of extensive research Can significantly increase data rates and reduce BER Telatar’s paper Bell Labs (1995)

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capacity of multi antenna gaussian channels i e telatar

MIT 6.441

Capacity of multi-antenna Gaussian Channels, I. E. Telatar

May 11, 2006

By: Imad Jabbour

introduction
Introduction
  • MIMO systems in wireless comm.
    • Recently subject of extensive research
    • Can significantly increase data rates and reduce BER
  • Telatar’s paper
    • Bell Labs (1995)
    • Information-theoretic aspect of single-user MIMO systems
    • Classical paper in the field

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

preliminaries
Preliminaries
  • Wireless fading scalar channel
    • DT Representation:
    • H is the complex channel fadingcoefficient
    • W is the complex noise,
    • Rayleigh fading: , such that |H| is Rayleigh distributed
  • Circularly-symmetric Gaussian
    • i.i.d. real and imaginary parts
    • Distribution invariant to rotations

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

mimo channel model 1
MIMO Channel Model (1)
  • I/O relationship
    • Design parameters
      • t Tx. antennas and r Rx. antennas
      • Fading matrix
      • Noise
    • Power constraint:
  • Assumption
    • H known at Rx. (CSIR)

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

mimo channel model 2
MIMO Channel Model (2)
  • System representation
  • Telatar: the fading matrix H can be
    • Deterministic
    • Random and changes over time
    • Random, but fixed once chosen

Transmitter

Receiver

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

deterministic fading channel 1
Deterministic Fading Channel (1)
  • Fading matrix is not random
    • Known to bothTx. and Rx.
    • Idea: Convert vector channel to a parallel one
  • Singular value decomposition of H
    • SVD: , for U and V unitary, and D diagonal
    • Equivalent system: , where
    • Entries of D are the singular values of H
      • There are singular values

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

deterministic fading channel 2
Deterministic Fading Channel (2)
  • Equivalent parallel channel [nmin=min(r,t)]
    • Tx. must know H to pre-process it, and Rx. must know H to post-process it

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

deterministic fading channel 3
Deterministic Fading Channel (3)
  • Result of SVD
    • Parallel channel with sub-channels
    • Water-filling maximizes capacity
    • Capacity is
      • Optimal power allocation
      • is chosen to meet total power constraint

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

random varying channel 1
Random Varying Channel (1)
  • Random channel matrix H
    • Independent of both X and W, and memoryless
    • Matrix entries
  • Fast fading
    • Channel varies much faster than delay requirement
    • Coherence time (Tc): period of variation of channel

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

random varying channel 2
Random Varying Channel (2)
  • Information-theoretic aspect
    • Codeword length should average out both additive noise and channel fluctuations
  • Assume that Rx. tracks channel perfectly
    • Capacity is
    • Equal power allocation at Tx.
    • Can show that
    • At high power, C scales linearly with nmin
    • Results also apply for any ergodic H

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

random varying channel 3
Random Varying Channel (3)
  • MIMO capacity versus SNR (from [2])

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

random fixed channel 1
Random Fixed Channel (1)
  • Slow fading
    • Channel varies much slower than delay requirement
    • H still random, but is constant over transmission duration of codeword
  • What is the capacity of this channel?
    • Non-zero probability that realization of H does not support the data rate
    • In this sense, capacity is zero!

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

random fixed channel 2
Random Fixed Channel (2)
  • Telatar’s solution: outage probability pout
    • pout is probability that R is greater that maximum achievable rate
    • Alternative performance measure is
      • Largest R for which
      • Optimal power allocation is equal allocation across only a subset of the Tx. antennas.

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

discussion and analysis 1
Discussion and Analysis (1)
  • What’s missing in the picture?
    • If H is unknown at Tx., cannot do SVD
      • Solution: V-BLAST
    • If H is known at Tx. also (full CSI)
      • Power gain over CSIR
    • If H is unknown at both Tx. and Rx (non-coherent model)
      • At high SNR, solution given by Marzetta & Hochwald, and Zheng
    • Receiver architectures to achieve capacity
    • Other open problems

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

discussion and analysis 2
Discussion and Analysis (2)
  • If H unknown at Tx.
    • Idea: multiplex in an arbitrary coordinate system B, and do joint ML decoding at Rx.
    • V-BLAST architecture can achieve capacity

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

discussion and analysis 3
Discussion and Analysis (3)
  • If varying H known at Tx. (full CSI)
    • Solution is now water-filling over space and time
    • Can show optimal power allocation is P/nmin
    • Capacity is
    • What are we gaining?
      • Power gain of nt/nmin as compared to CSIR case

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

discussion and analysis 4
Discussion and Analysis (4)
  • If H unknown at both Rx. and Tx.
    • Non-coherent channel: channel changes very quickly so that Rx. can no more track it
    • Block fading model
    • At high SNR, capacity gain is equal to (Zheng)

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

discussion and analysis 5
Discussion and Analysis (5)
  • Receiver architectures [2]
    • V-BLAST can achieve capacity for fast Rayleigh-fading channels
    • Caveat: Complexity of joint decoding
    • Solution: simpler linear decoders
      • Zero-forcing receiver (decorrelator)
      • MMSE receiver
      • MMSE can achieve capacity if SIC is used

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

discussion and analysis 6
Discussion and Analysis (6)
  • Open research topics
    • Alternative fading models
    • Diversity/multiplexing tradeoff (Zheng & Tse)
  • Conclusion
    • MIMO can greatly increase capacity
    • For coherent high SNR,
    • How many antennas are we using?
    • Can we “beat” the AWGN capacity?

MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

thank you

Thank you!

Any questions?